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Introduction to Integrals

The Area as a Sum

The Definite Integral

Problem :
Consider the function
f (x) = x2 + 1
on the interval
[0, 2]
.
Using four subdivisions, find the left-hand approximation,
L4
, of the
area under the curve of
f
on the interval indicated.

Δx

= = =

L4

= f (0) + f () + f (1) + f ()

= 1 + +2 +

= =

Problem :
For the same function, using four subdivisions, find the right-hand sum,
R4
.

R4

= f () + f (1) + f () + f (2)

= +2 + + 5

= =

Problem :
For the same function, using four subdivisions, find the midpoint sum,
M4
.

M4

= f () + f () + f () + f ()

= + + +

= =

Notice that on the interval in question,
f
is a strictly increasing function. If
f
is
increasing on an interval, then
Ln < Mn < Rn
. If
f
is decreasing on an interval,
then
Rn < Mn < Ln
.

Problem :
Find

f (xk)Δx for f (x) = 2x on [0, 2]

To solve this problem, notice that the graph of
f (x)
is a line, and the
area in question is in the shape of a right triangle with base 2 and
height
f (2) = 4
. So, the limit of the right hand sum, which
is the area under the curve, is

(2)(4) = 4

Problem :
Find

f (xk)Δx for f (x) = on [0, 3]

To solve this problem, notice that the graph of
f (x)
is a semicircle of
radius 3 centered at the origin. The interval
[0, 3]
contains a section
of the curve that is equivalent to a
quarter-circle. Thus, the area under the curve is equal to one-fourth the
area of a circle
with radius 3, or
Π(32) = Π
.